The Black Cats remain interested in Broadhead after a successful loan spell at the Stadium of Light but look set to face competition from Championship rivals.
It has been reported by The Athletic that Everton have offered Broadhead a new deal and plan to loan him out again this summer.
The 24-year-old doesn’t appear to be in Lampard’s first-team plans despite an injury to Dominic Calvert-Lewin.
Supercomputer predicts final Championship table - including surprise finishes for Sunderland, Middlesbrough, QPR, Sheffield United and co
Sunderland AFC transfer news: Cats take 'brilliant' ex-Manchester United and Burnley midfielder on trial
Championship transfer news: Interest in £22million man, setback for Arsenal loanee plus Norwich reports
Sunderland AFC transfer gossip: Cats in 'final stages' of deal for 18-y/o eight-cap international winger
Sheffield United suffer injury blow ahead of Sunderland game with 264 goal striker potentially sidelined
And against Chelsea, Lampard opted to play Alex Iwobi, Dwight McNeil and Anthony Gordon as his forwards, without a recognised natural number nine.
In his pre-match press conference ahead of their game against Chelsea, Lampard even spoke about the possibility of playing midfielder Dele Ali up front.
"The first thing I'll say is I don't think we're the only team still looking for solutions,” said Lampard. “When the window goes three weeks into the season, to say everyone needs to get their business done early sounds great, but it's not always the way. Some have talked about younger players going on loan, they need development."
Sunderland and Everton have already done business this summer with striker Ellis Simms arriving earlier in the window. The 21-year-old made an instant impact for Sunderland, scoring a brace on his full debut against Bristol City.
EFL rules state that no club may sign more than four players on loan from another club during the season. No more than two of those players may be over the age of 23, meaning Sunderland can still sign Broadhead on loan.